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On reducing submodules of Hilbert modules with G(n)-invariant kernels

Biswas, Shibananda and Ghosh, Gargi and Misra, Gadadhar and Roy, Subrata Shyam (2019) On reducing submodules of Hilbert modules with G(n)-invariant kernels. In: JOURNAL OF FUNCTIONAL ANALYSIS, 276 (3). pp. 751-784.

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Official URL: https://doi.org/10.1016/j.jfa.2018.10.025

Abstract

Fix a bounded domain Omega subset of C-n and a positive definite kernel K on Omega, both invariant under G(n), the permutation group on n symbols. Let H subset of Hol(Omega) be the Hilbert module determined by K. We show that H splits into orthogonal direct sum of subspaces PpH indexed by the partitions p of n. We prove that each submodule PpH is a locally free Hilbert module of rank equal to square of the dimension chi(p) (1) of the irreducible representation corresponding to p. Given two partitions p and q, we show that if chi(p)(1) not equal chi(q)(1), then the sub-modules PpH and PqH are not unitarily equivalent. We prove that if H is a contractive analytic Hilbert module on Omega, then the `Taylor joint spectrum of the n-tuple of multiplication operators by elementary symmetric polynomials on PpH is clos (s(Omega)), where s : C-n -> C-n is the symmetrization map. It is then shown that this commuting tuple of operators defines a contractive homomorphism of the ring of symmetric polynomials Cz](Gn) in n variables, equipped with the sup norm on clos (s(Omega)). (C) 2018 Elsevier Inc. All rights reserved.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to ACADEMIC PRESS INC ELSEVIER SCIENCE
Keywords: Reducing submodules; G(n)-invariant reproducing kernel; Symmetrized polydisc; Weighted Bergman space
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 28 Jan 2019 08:56
Last Modified: 28 Jan 2019 08:56
URI: http://eprints.iisc.ac.in/id/eprint/61474

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